Normalized defining polynomial
\( x^{11} - 33x^{9} + 396x^{7} - 2079x^{5} + 4455x^{3} - 2673x - 837 \)
Invariants
| Degree: | $11$ |
| |
| Signature: | $(11, 0)$ |
| |
| Discriminant: |
\(2713285598714072534889\)
\(\medspace = 3^{10}\cdot 11^{16}\)
|
| |
| Root discriminant: | \(88.82\) |
| |
| Galois root discriminant: | $3^{10/11}11^{84/55}\approx 105.73949735950434$ | ||
| Ramified primes: |
\(3\), \(11\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3}a^{4}$, $\frac{1}{3}a^{5}$, $\frac{1}{3}a^{6}$, $\frac{1}{3}a^{7}$, $\frac{1}{9}a^{8}$, $\frac{1}{9}a^{9}$, $\frac{1}{9}a^{10}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
| |
| Narrow class group: | $C_{2}\times C_{2}$, which has order $4$ |
|
Unit group
| Rank: | $10$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{9}a^{9}-\frac{1}{9}a^{8}-3a^{7}+\frac{11}{3}a^{6}+\frac{76}{3}a^{5}-\frac{115}{3}a^{4}-63a^{3}+128a^{2}-12a-47$, $\frac{1}{9}a^{9}+\frac{2}{9}a^{8}-3a^{7}-\frac{17}{3}a^{6}+\frac{83}{3}a^{5}+\frac{139}{3}a^{4}-103a^{3}-132a^{2}+138a+91$, $\frac{1}{9}a^{8}-\frac{1}{3}a^{7}-\frac{7}{3}a^{6}+\frac{19}{3}a^{5}+\frac{47}{3}a^{4}-34a^{3}-41a^{2}+50a+29$, $\frac{2}{9}a^{10}+\frac{5}{9}a^{9}-\frac{50}{9}a^{8}-\frac{40}{3}a^{7}+\frac{134}{3}a^{6}+\frac{295}{3}a^{5}-132a^{4}-232a^{3}+147a^{2}+132a+8$, $\frac{1}{9}a^{8}-\frac{8}{3}a^{6}+20a^{4}-2a^{3}-48a^{2}+17a+17$, $\frac{1}{9}a^{10}-\frac{1}{9}a^{9}-\frac{28}{9}a^{8}+\frac{8}{3}a^{7}+31a^{6}-\frac{67}{3}a^{5}-\frac{397}{3}a^{4}+80a^{3}+221a^{2}-110a-85$, $\frac{4}{9}a^{10}+\frac{7}{9}a^{9}-\frac{127}{9}a^{8}-\frac{68}{3}a^{7}+154a^{6}+\frac{694}{3}a^{5}-647a^{4}-936a^{3}+718a^{2}+1018a+256$, $\frac{1}{9}a^{10}+\frac{2}{3}a^{9}-\frac{25}{9}a^{8}-16a^{7}+\frac{64}{3}a^{6}+121a^{5}-54a^{4}-311a^{3}+45a^{2}+209a+61$, $\frac{1}{3}a^{10}+a^{9}-8a^{8}-\frac{73}{3}a^{7}+\frac{176}{3}a^{6}+\frac{550}{3}a^{5}-\frac{404}{3}a^{4}-450a^{3}+79a^{2}+303a+88$, $\frac{1}{9}a^{9}-\frac{11}{3}a^{7}+a^{6}+\frac{118}{3}a^{5}-\frac{44}{3}a^{4}-149a^{3}+35a^{2}+133a+41$
|
| |
| Regulator: | \( 48372191.2791 \) |
| |
| Unit signature rank: | \( 9 \) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{11}\cdot(2\pi)^{0}\cdot 48372191.2791 \cdot 1}{2\cdot\sqrt{2713285598714072534889}}\cr\approx \mathstrut & 0.95092871811 \end{aligned}\]
Galois group
$C_{11}:C_5$ (as 11T3):
| A solvable group of order 55 |
| The 7 conjugacy class representatives for $C_{11}:C_5$ |
| Character table for $C_{11}:C_5$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.5.0.1}{5} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} }$ | R | ${\href{/padicField/5.5.0.1}{5} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | R | ${\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.5.0.1}{5} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.11.0.1}{11} }$ | ${\href{/padicField/29.5.0.1}{5} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.11.0.1}{11} }$ | ${\href{/padicField/47.5.0.1}{5} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.1.11.10a1.1 | $x^{11} + 3$ | $11$ | $1$ | $10$ | $C_{11}:C_5$ | $$[\ ]_{11}^{5}$$ |
|
\(11\)
| 11.1.11.16a2.1 | $x^{11} + 22 x^{6} + 11$ | $11$ | $1$ | $16$ | $C_{11}:C_5$ | $$[\frac{8}{5}]_{5}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *55 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.11.5t1.a.a | $1$ | $ 11 $ | \(\Q(\zeta_{11})^+\) | $C_5$ (as 5T1) | $0$ | $1$ | |
| 1.11.5t1.a.b | $1$ | $ 11 $ | \(\Q(\zeta_{11})^+\) | $C_5$ (as 5T1) | $0$ | $1$ | |
| 1.11.5t1.a.c | $1$ | $ 11 $ | \(\Q(\zeta_{11})^+\) | $C_5$ (as 5T1) | $0$ | $1$ | |
| 1.11.5t1.a.d | $1$ | $ 11 $ | \(\Q(\zeta_{11})^+\) | $C_5$ (as 5T1) | $0$ | $1$ | |
| *55 | 5.52089208083.11t3.a.a | $5$ | $ 3^{5} \cdot 11^{8}$ | 11.11.2713285598714072534889.1 | $C_{11}:C_5$ (as 11T3) | $0$ | $5$ |
| *55 | 5.52089208083.11t3.a.b | $5$ | $ 3^{5} \cdot 11^{8}$ | 11.11.2713285598714072534889.1 | $C_{11}:C_5$ (as 11T3) | $0$ | $5$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.